The best approximation and the values of the widths of some classes of analytical functions in the weighted Bergman space B_(2,γ)

Abstract

In the paper, exact inequalities are found for the best approximation of an arbitrary analytic function f in the unit circle by algebraic complex polynomials in terms of the modulus of continuity of the m th order of the r th order derivative f^((r)) in the weighted Bergman space B_(2,γ). Also using the modulus of continuity of the m-th order of the derivative f(r), we introduce a class of functions W_m^((r) ) (h,Φ) analytic in the unit circle and defined by a given majorant Φ, h∈(0,π⁄n], n>r, monotonically increasing on the positive semiaxis. Under certain conditions on the majorant Φ, for the introduced class of functions, the exact values of some known n-widths are calculated. We use methods for solving extremal problems in normed spaces of functions analytic in a circle, as well as the method for estimating from below the n-widths of functional classes in various Banach spaces developed by V.M. Tikhomirov. The results presented in this paper are a continuation and generalization of some earlier results on the best approximations and values of widths in the weighted Bergman space B_(2,γ).